gsl_nls_loss: Robust loss functions with tunable parameters
In gslnls: GSL Multi-Start Nonlinear Least-Squares Fitting

gsl_nls_loss

R Documentation

Robust loss functions with tunable parameters

Description

Allow the user to tune the coefficient(s) of the robust loss functions supported by gsl_nls. For all choices other than rho = "default", the MM-estimation problem is optimized by means of iterative reweighted least squares (IRLS).

Usage

gsl_nls_loss(
  rho = c("default", "huber", "barron", "bisquare", "welsh", "optimal", "hampel", "ggw",
    "lqq"),
  cc = NULL
)

Arguments

`rho`	character loss function, one of `"default", "huber", "barron", "bisquare", "welsh", "optimal", "hampel", "ggw", "lqq"`.
`cc`	named or unnamed numeric vector of tuning parameters. The length of this argument depends on the selected `rho` function (see ‘Details’). If `NULL`, the default tuning parameters are returned.

Value

A list with two components:

with meanings as explained under ‘Arguments’.

Loss function details

default

Default squared loss, no iterative reweighted least squares (IRLS) is required in this case.

\rho(x) = x^2

huber

Huber loss function with scaling constant k, defaulting to k = 1.345 for 95% efficiency of the regression estimator.

\rho(x, k) = \left\{ \begin{array}{ll} \frac{1}{2} x^2 &\quad \text{ if } |x| \leq k \\[2pt] k(|x| - \frac{k}{2}) &\quad \text{ if } |x| > k \end{array} \right.

barron

Barron's smooth family of loss functions with robustness parameter \alpha \leq 2 (default \alpha = 1) and scaling constant k (default k = 1.345). Special cases include: (scaled) squared loss for \alpha = 2; L1-L2 loss for \alpha = 1; Cauchy loss for \alpha = 0; Geman-McClure loss for \alpha = -2; and Welsch/Leclerc loss for \alpha = -\infty. See Barron (2019) for additional details.

\rho(x, \alpha, k) = \left\{ \begin{array}{ll} \frac{1}{2} (x / k)^2 &\quad \text{ if } \alpha = 2 \\[2pt] \log\left(\frac{1}{2} (x / k)^2 + 1 \right) &\quad \text{ if } \alpha = 0 \\[2pt] 1 - \exp\left( -\frac{1}{2} (x / k)^2 \right) &\quad \text{ if } \alpha = -\infty \\[2pt] \frac{|\alpha - 2|}{\alpha} \left( \left( \frac{(x / k)^2}{|\alpha-2|} + 1 \right)^{\alpha / 2} - 1 \right) &\quad \text{ otherwise } \end{array} \right.

bisquare

Tukey's biweight/bisquare loss function with scaling constant k, defaulting to k = 4.685 for 95% efficiency of the regression estimator, (k = 1.548 gives a breakdown point of 0.5 of the S-estimator).

\rho(x, k) = \left\{ \begin{array}{ll} 1 - (1 - (x / k)^2)^3 &\quad \text{ if } |x| \leq k \\[2pt] 1 &\quad \text{ if } |x| > k \end{array} \right.

welsh

Welsh/Leclerc loss function with scaling constant k, defaulting to k = 2.11 for 95% efficiency of the regression estimator, (k = 0.577 gives a breakdown point of 0.5 of the S-estimator). This is equivalent to the Barron loss function with robustness parameter \alpha = -\infty.

\rho(x, k) = 1 - \exp\left( -\frac{1}{2} (x / k)^2 \right)

optimal

Optimal loss function as in Section 5.9.1 of Maronna et al. (2006) with scaling constant k, defaulting to k = 1.060 for 95% efficiency of the regression estimator, (k = 0.405 gives a breakdown point of 0.5 of the S-estimator).

\rho(x, k) = \int_0^x \text{sign}(u) \left( - \dfrac{\phi'(|u|) + k}{\phi(|u|)} \right)^+ du

with \phi the standard normal density.

hampel

Hampel loss function with a single scaling constant k, setting a = 1.5k, b = 3.5k and r = 8k. k = 0.902 by default, resulting in 95% efficiency of the regression estimator, (k = 0.212 gives a breakdown point of 0.5 of the S-estimator).

\rho(x, a, b, r) = \left\{ \begin{array}{ll} \frac{1}{2} x^2 / C &\quad \text{ if } |x| \leq a \\[2pt] \left( \frac{1}{2}a^2 + a(|x|-a)\right) / C &\quad \text{ if } a < |x| \leq b \\[2pt] \frac{a}{2}\left( 2b - a + (|x| - b) \left(1 + \frac{r - |x|}{r-b}\right) \right) / C &\quad \text{ if } b < |x| \leq r \\[2pt] 1 &\quad \text{ if } r < |x| \end{array} \right.

with C = \rho(\infty) = \frac{a}{2} (b - a + r).

ggw

Generalized Gauss-Weight loss function, a generalization of the Welsh/Leclerc loss with tuning constants a, b, c, defaulting to a = 1.387, b = 1.5, c = 1.063 for 95% efficiency of the regression estimator, (a = 0.204, b = 1.5, c = 0.296 gives a breakdown point of 0.5 of the S-estimator).

\rho(x, a, b, c) = \int_0^x \psi(u, a, b, c) du

with,

\psi(x, a, b, c) = \left\{ \begin{array}{ll} x &\quad \text{ if } |x| \leq c \\[2pt] \exp\left( -\frac{1}{2} \frac{(|x| - c)^b}{a} \right) x &\quad \text{ if } |x| > c \end{array} \right.

lqq

Linear Quadratic Quadratic loss function with tuning constants b, c, s, defaulting to b = 1.473, c = 0.982, s = 1.5 for 95% efficiency of the regression estimator, (b = 0.402, c = 0.268, s = 1.5 gives a breakdown point of 0.5 of the S-estimator).

\rho(x, b, c, s) = \int_0^x \psi(u, b, c, s) du

with,

\psi(x, b, c, s) = \left\{ \begin{array}{ll} x &\quad \text{ if } |x| \leq c \\[2pt] \text{sign}(x) \left( |x| - \frac{s}{2b} (|x| - c)^2 \right) &\quad \text{ if } c < |x| \leq b + c \\[2pt] \text{sign}(x) \left( c + b - \frac{bs}{2} + \frac{s-1}{a} \left( \frac{1}{2} \tilde{x}^2 - a\tilde{x}\right) \right) &\quad \text{ if } b + c < |x| \leq a + b + c \\[2pt] 0 &\quad \text{ otherwise } \end{array} \right.

where \tilde{x} = |x| - b - c and a = (2c + 2b - bs) / (s - 1).

References

J.T. Barron (2019). A general and adaptive robust loss function. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition (pp. 4331-4339).

M. Galassi et al., GNU Scientific Library Reference Manual (3rd Ed.), ISBN 0954612078.

R.A. Maronna et al., Robust Statistics: Theory and Methods, ISBN 0470010924.

Examples

## Huber loss with default scale k
gsl_nls_loss(rho = "huber")

## L1-L2 loss (alpha = 1)
gsl_nls_loss(rho = "barron", cc = c(1, 1.345))

## Cauchy loss (alpha = 0)
gsl_nls_loss(rho = "barron", cc = c(k = 1.345, alpha = 0))

gslnls documentation built on April 3, 2025, 8:59 p.m.